Isometry group of the Poincare half-plane
= Isometry group of the Poincare half-plane
{c}
{title2=$\operatorname{Isom}(\mathbb H^2)$}
Every orientation-preserving isometry of the <Poincaré half-plane model> lies in $PSL_2(\mathbb R)$. Every orientation-reversing isometry is the composition of one of these with the <hyperbolic reflection> $z\mapsto-\overline z$ in the imaginary axis. Equivalently, the full isometry group consists of real Möbius and anti-Möbius transformations, modulo nonzero scalar matrices.